<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:63"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<i><font color="Green">E33</font></i>: <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E3:63"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:63"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:63"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 ;<br/>


</div>
